Bubble Dynamics and Shock Waves pp 141-175

Part of the Shock Wave Science and Technology Reference Library book series (SHOCKWAVES, volume 8)

Shock Propagation in Polydisperse Bubbly Liquids

  • Keita Ando
  • Tim Colonius
  • Christopher E. Brennen

Abstract

We investigate the shock dynamics of liquid flows containing small gas bubbles with numerical simulations based on a continuum bubbly flow model. Particular attention is devoted to the effects of distributed bubble sizes and gas-phase nonlinearity on shock dynamics. Ensemble-averaged conservation laws for polydisperse bubbly flows are closed with a Rayleigh–Plesset-type model for single bubble dynamics. Numerical simulations of one-dimensional shock propagation reveal that phase cancellations in the oscillations of different-sized bubbles can lead to an apparent damping of the averaged shock dynamics. Experimentally, we study the propagation of waves in a deformable tube filled with a bubbly liquid. The model is extended to quasi-one-dimensional cases. This leads to steady shock relations that account for the compressibility associated with tube deformation, bubbles and host liquid. A comparison between the theory and the water-hammer experiments suggests that the gas-phase nonlinearity plays an essential role in the propagation of shocks.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Delale, C.F., Nas, S., Tryggvason, G.: Direct numerical simulations of shock propagation in bubbly liquids. Phys. Fluids 17, Art. No. 121705 (2005)Google Scholar
  2. 2.
    Delale, C.F., Tryggvason, G.: Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations. Shock Waves 17, 433–440 (2008)CrossRefGoogle Scholar
  3. 3.
    Lu, T., Samulyak, R., Glimm, J.: Direct numerical simulation of bubbly flows and application to cavitation mitigation. J. Fluids Eng. 129, 595–604 (2007)CrossRefGoogle Scholar
  4. 4.
    Seo, J.H., Lele, S.K., Tryggvason, G.: Investigation and modeling of bubble-bubble interaction effect in homogeneous bubbly flows. Phys. Fluids 22, Art. No. 063302 (2010)Google Scholar
  5. 5.
    Arndt, R.E.A.: Cavitation in fluid machinery and hydraulic structures. Annu. Rev. Fluid Mech. 13, 273–328 (1981)CrossRefGoogle Scholar
  6. 6.
    Brennen, C.E.: Hydrodynamics of Pumps. Oxford University Press (1994)Google Scholar
  7. 7.
    Cole, R.H.: Underwater Explosions. Princeton University Press (1948)Google Scholar
  8. 8.
    Kedrinskii, V.K.: Hydrodynamics of Explosion. Springer (2005)Google Scholar
  9. 9.
    Bailey, M.R., McAteer, J.A., Pishchalnikov, Y.A., Hamilton, M.F., Colonius, T.: Progress in lithotripsy research. Acoust. Today 18, 18–29 (2006)CrossRefGoogle Scholar
  10. 10.
    Krimmel, J., Colonius, T., Tanguay, M.: Simulation of the effects of cavitation and anatomy in the shock path of model lithotripters. Urol. Res. 38, 505–518 (2010)CrossRefGoogle Scholar
  11. 11.
    Brennen, C.E.: Cavitation and Bubble Dynamics. Oxford University Press (1995)Google Scholar
  12. 12.
    Brennen, C.E.: Fundamentals of Multiphase Flow. Cambridge University Press (2005)Google Scholar
  13. 13.
    Commander, K.W., Prosperetti, A.: Linear pressure waves in bubbly liquids: Comparison between theory and experiments. J. Acoust. Soc. Am. 85, 732–746 (1989)CrossRefGoogle Scholar
  14. 14.
    Nigmatulin, R.I.: Mathematical modelling of bubbly liquid motion and hydrodynamical effects in wave propagation phenomenon. Appl. Sci. Res. 38, 267–289 (1982)MATHCrossRefGoogle Scholar
  15. 15.
    van Wijngaarden, L.: One-dimensional flow of liquids containing small gas bubbles. Annu. Rev. Fluid Mech. 4, 369–396 (1972)CrossRefGoogle Scholar
  16. 16.
    Campbell, I.J., Pitcher, A.S.: Shock waves in a liquid containing gas bubbles. Proc. R Soc. Lond. A 243, 534–545 (1958)MATHCrossRefGoogle Scholar
  17. 17.
    Beylich, A.E., Gülhan, A.: On the structure of nonlinear waves in liquids with gas bubbles. Phys. Fluids A 2, 1412–1428 (1990)MATHCrossRefGoogle Scholar
  18. 18.
    Kameda, M., Matsumoto, Y.: Shock waves in a liquid containing small gas bubbles. Phys. Fluids 8, 322–335 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kameda, M., Shimaura, N., Higashino, F., Matsumoto, Y.: Shock waves in a uniform bubbly flow. Phys. Fluids 10, 2661–2668 (1998)CrossRefGoogle Scholar
  20. 20.
    Noordzij, L., van Wijngaarden, L.: Relaxation effects, caused by relative motion, on shock waves in gas-bubble/liquid mixtures. J. Fluid Mech. 66, 115–143 (1974)MATHCrossRefGoogle Scholar
  21. 21.
    Tan, M.J., Bankoff, S.G.: Strong shock waves propagating through a bubbly mixture. Exp. Fluids 2, 159–165 (1984)CrossRefGoogle Scholar
  22. 22.
    Nigmatulin, R.I., Khabeev, N.S., Hai, Z.N.: Waves in liquids with vapour bubbles. J. Fluid Mech. 186, 85–117 (1988)MATHCrossRefGoogle Scholar
  23. 23.
    Tan, M.J., Bankoff, S.G.: Propagation of pressure waves in bubbly mixtures. Phys. Fluids 27, 1362–1369 (1984)MATHCrossRefGoogle Scholar
  24. 24.
    Watanabe, M., Prosperetti, A.: Shock waves in dilute bubbly liquids. J. Fluid Mech. 274, 349–381 (1994)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Brennen, C.E.: Fission of collapsing cavitation bubbles. J. Fluid Mech. 472, 153–166 (2002)MATHCrossRefGoogle Scholar
  26. 26.
    Shepherd, J.E., Inaba, K.: Shock loading and failure of fluid-filled tubular structures. In: Shukla, A., Ravichandran, G., Rajapakse, Y.D.S. (eds.) Dynamic Failure of Materials and Structures, pp. 153–190. Springer (2010)Google Scholar
  27. 27.
    Tijsseling, A.S.: Fluid-structure interaction in liquid-filled pipe systems: A review. J. Fluids Struct. 10, 109–146 (1996)CrossRefGoogle Scholar
  28. 28.
    Wylie, E.B., Streeter, V.L.: Fluid Transients in Systems. Prentice Hall (1993)Google Scholar
  29. 29.
    Joukowsky, N.E.: Memoirs of the Imperial Academy Society of St. Petersburg. Proc. Amer. Water Works Assoc. 24, 341–424 (1898)Google Scholar
  30. 30.
    Korteweg, D.J.: Ber die Fortpflanzungsgeschwindigkeit des Shalles in elastischen Röhren. Annalen der Physik und Chemie. 5, 525–542 (1878)MATHGoogle Scholar
  31. 31.
    Kobori, T., Yokoyama, S., Miyashiro, H.: Propagation velocity of pressure wave in pipe line. Hitachi Hyoron. 37, 33–37 (1955)Google Scholar
  32. 32.
    Dashpande, V.S., Heaver, A., Fleck, N.A.: An underwater shock simulator. Proc. R. Soc. A 462, 1021–1041 (2006)CrossRefGoogle Scholar
  33. 33.
    Espinosa, H.D., Lee, S., Moldovan, N.: A novel fluid structure interaction experiment to investigate deformation of structural elements subjected to impulsive loading. Exp. Mech. 46, 805–824 (2006)CrossRefGoogle Scholar
  34. 34.
    Inaba, K., Shepherd, J.E.: Flexural waves in fluid-filled tubes subject to axial impact. J. Pressure Vessel Technol. 132, Art. No. 021302 (2010)Google Scholar
  35. 35.
    Ando, K., Sanada, T., Inaba, K., Damazo, J.S., Shepherd, J.E., Colonius, T., Brennen, C.E.: Shock propagation through a bubbly liquid in a deformable tube. J. Fluid Mech. 671, 339–363 (2011)MATHCrossRefGoogle Scholar
  36. 36.
    Ando, K., Colonius, T., Brennen, C.E.: Numerical simulation of shock propagation in a polydisperse bubbly liquid. Int. J. Multiphase Flow 37, 596–608 (2011)CrossRefGoogle Scholar
  37. 37.
    Ishii, M., Hibiki, T.: Thermo-Fluid Dynamics of Two-Phase Flow. Springer (2006)Google Scholar
  38. 38.
    Zhang, Z.D., Prosperetti, A.: Ensemble-averaged equations for bubbly flows. Phys. Fluids 6, 2956–2970 (1994)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Batchelor, G.K.: The stress system in a suspension of force-free particles. J. Fluid Mech. 41, 545–570 (1970)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Biesheuvel, A., van Wijngaarden, L.: Two-phase flow equations for a dilute dispersion of gas bubbles in liquid. J. Fluid Mech. 148, 301–318 (1984)MATHCrossRefGoogle Scholar
  41. 41.
    Nigmatulin, R.I.: Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int. J. Heat Mass Transfer 5, 353–385 (1979)MATHGoogle Scholar
  42. 42.
    Prosperetti, A.: Fundamental acoustic properties of bubbly liquids. In: Levy, M., Bass, H.E., Stern, R.R. (eds.) Handbook of Elastic Properties of Solids, Liquids, and Gases. Elastic Properties of Fluids: Liquids and Gases, vol. 4, pp. 183–205. Academic (2001)Google Scholar
  43. 43.
    Zhang, Z.D., Prosperetti, A.: Averaged equations for inviscid disperse two-phase flow. J. Fluid Mech. 267, 185–219 (1994)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Zhang, Z.D., Prosperetti, A.: Momentum and energy equations for disperse two-phase flows and their closure for dilute suspensions. Int. J. Multiphase Flow 23, 425–453 (1997)MATHCrossRefGoogle Scholar
  45. 45.
    Caflisch, R.E., Miksis, M.J., Papanicolaou, G.C., Ting, L.: Wave propagation in bubbly liquids at finite volume fraction. J. Fluid Mech. 160, 1–14 (1985)MATHCrossRefGoogle Scholar
  46. 46.
    Fuster, D., Colonius, T.: Modelling bubble clusters in compressible liquids. J. Fluid Mech. 688, 352–389 (2011)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Thompson, P.A.: Compressible Fluid Dynamics. McGraw-Hill (1972)Google Scholar
  48. 48.
    Van Wijngaarden, L.: On the equations of motion for mixtures of liquid and gas bubbles. J. Fluid Mech. 33, 465–474 (1968)MATHCrossRefGoogle Scholar
  49. 49.
    Epstein, P.S., Plesset, M.S.: On the stability of gas bubbles in liquid-gas solutions. J. Chem. Phys. 18, 1505–1509 (1950)CrossRefGoogle Scholar
  50. 50.
    Plesset, M.S., Prosperetti, A.: Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9, 145–185 (1977)CrossRefGoogle Scholar
  51. 51.
    Colonius, T., Hagmeijer, J., Ando, K., Brennen, C.E.: Statistical equilibrium of bubble oscillations in dilute bubbly flows. Phys. Fluids 20, Art. No. 040902 (2008)Google Scholar
  52. 52.
    Delale, C.F., Tunç, M.: A bubble fission model for collapsing cavitation bubbles. Phys. Fluids 16, 4200–4203 (2004)CrossRefGoogle Scholar
  53. 53.
    Johnsen, E., Colonius, T.: Numerical simulations of non-spherical bubble collapse. J. Fluid Mech. 629, 231–262 (2009)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Ranjan, D., Oakley, J., Bonazza, R.: Shock-bubble interactions. Annu. Rev. Fluid Mech. 43, 117–140 (2011)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Gilmore, F.R.: The collapse and growth of a spherical bubble in a viscous compressible liquid. California Institute of Technology. Hydrodynamics Laboratory Report No. 26–4 (1952)Google Scholar
  56. 56.
    Plesset, M.S.: The dynamics of cavitation bubbles. J. Appl. Mech. 16, 228–231 (1949)Google Scholar
  57. 57.
    Rayleigh, L.: On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 34, 94–98 (1917)MATHCrossRefGoogle Scholar
  58. 58.
    Caflisch, R.E., Miksis, M.J., Papanicolaou, G.C., Ting, L.: Effective equations for wave propagation in bubbly liquids. J. Fluid Mech. 153, 259–273 (1985)MATHCrossRefGoogle Scholar
  59. 59.
    Takahira, H.: A remark on the pressure terms in the Rayleigh-Plesset equation for cavitating flows. Trans. Jpn. Soc. Mech. Eng. B 70, 617–622 (2004)CrossRefGoogle Scholar
  60. 60.
    Hao, Y., Prosperetti, A.: The dynamics of vapor bubbles in acoustic pressure fields. Phys. Fluids 11, 2008–2019 (1999)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Preston, A., Colonius, T., Brennen, C.E.: Toward efficient computation of heat and mass transfer effects in the continuum model for bubbly cavitating flows. In: Proceedings of the Fourth International Symposium on Cavitation (2001)Google Scholar
  62. 62.
    Lin, H., Storey, B.D., Szeri, A.J.: Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation. J. Fluid Mech. 452, 145–162 (2002)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Prosperetti, A., Crum, L.A., Commander, K.W.: Nonlinear bubble dynamics. J. Acoust. Soc. Am. 83, 502–514 (1988)CrossRefGoogle Scholar
  64. 64.
    Nigmatulin, R.I., Khabeev, N.S., Nagiev, F.B.: Dynamics, heat and mass transfer of vapour-gas bubbles in a liquid. Int. J. Heat Mass Transfer 24, 1033–1044 (1981)MATHCrossRefGoogle Scholar
  65. 65.
    Fujikawa, S., Akamatsu, T.: Effects of the non-equilibrium condensation of vapour on the pressure wave produced by the collapse of a bubble in a liquid. J. Fluid Mech. 97, 481–512 (1980)MATHCrossRefGoogle Scholar
  66. 66.
    Preston, A.T., Colonius, T., Brennen, C.E.: A reduced-order model of diffusive effects on the dynamics of bubbles. Phys. Fluids 19, Art. No. 123302 (2007)Google Scholar
  67. 67.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser Verlag (1992)Google Scholar
  68. 68.
    Carstensen, E.L., Foldy, L.L.: Propagation of sound through a liquid containing bubbles. J. Acoust. Soc. Am. 19, 481–501 (1947)CrossRefGoogle Scholar
  69. 69.
    Foldy, L.L.: The muntiple scattering of waves. Phys. Rev. 67, 107–119 (1945)MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Prosperetti, A.: Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 61, 17–27 (1977)CrossRefGoogle Scholar
  71. 71.
    Ainslie, M.A., Leighton, T.G.: Review of scattering and extinction cross-sections, damping factors, and resonance frequencies of a spherical gas bubble. J. Acoust. Soc. Am. 130, 3184–3208 (2011)CrossRefGoogle Scholar
  72. 72.
    Minnaert, M.: On musical air-bubbles and sounds of running water. Phil. Mag. 16, 235–248 (1933)Google Scholar
  73. 73.
    Waterman, P.C., Truell, R.: Multiple scattering of waves. J. Math. Phys. 2, 512–537 (1961)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    Feuillade, C.: The attenuation and dispersion of sound in water containing multiply interacting air bubbles. J. Acoust. Soc. Am. 99, 3412–3430 (1996)CrossRefGoogle Scholar
  75. 75.
    Ando, K., Colonius, T., Brennen, C.E.: Improvement of acoustic theory of ultrasonic waves in dilute bubbly liquids. J. Acoust. Soc. Am. 126, EL69–EL74 (2009)Google Scholar
  76. 76.
    Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill (1978)Google Scholar
  77. 77.
    Smereka, P.: A Vlasov equation for pressure wave propagation in bubbly fluids. J. Fluid Mech. 454, 287–325 (2002)MathSciNetMATHCrossRefGoogle Scholar
  78. 78.
    Liu, X.-D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115, 200–212 (1994)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. NASA Langley Research Center ICASE Report No. 97–65 (1997)Google Scholar
  80. 80.
    Balsara, D.S., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)MathSciNetMATHCrossRefGoogle Scholar
  81. 81.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231–303 (1987)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Qiu, J., Shu, C.-W.: On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes. J. Comput. Phys. 183, 187–209 (2002)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer (2009)Google Scholar
  84. 84.
    Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)MATHCrossRefGoogle Scholar
  85. 85.
    Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25, 35–61 (1983)MathSciNetMATHCrossRefGoogle Scholar
  86. 86.
    Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Compt. 67, 73–85 (1998)MathSciNetMATHCrossRefGoogle Scholar
  87. 87.
    Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)MathSciNetMATHCrossRefGoogle Scholar
  88. 88.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002)Google Scholar
  89. 89.
    Tanguay, M.: Computation of bubbly cavitating flow in shock wave lithotripsy. PhD Thesis. California Institute of Technology (2004), http://thesis.library.caltech.edu/2188/
  90. 90.
    Colonius, T., d’Auria, F., Brennen, C.E.: Acoustic saturation in bubbly cavitating flow adjacent to an oscillating wall. Phys. Fluids 12, 2752–2761 (2000)CrossRefGoogle Scholar
  91. 91.
    Preston, A.T., Colonius, T., Brennen, C.E.: A numerical investigation of unsteady bubbly cavitating nozzle flows. Phys. Fluids 14, 300–311 (2002)CrossRefGoogle Scholar
  92. 92.
    Skalak, R.: An extension of the theory of water hammer. Trans. ASME 78, 105–116 (1956)Google Scholar
  93. 93.
    Tijsseling, A.S., Lambert, M.F., Simpson, A.R., Stephens, M.L., Vítkovský, J.P., Bergant, A.: Skalak’s extended theory of water hammer. J. Sound Vib. 310, 718–728 (2008)CrossRefGoogle Scholar
  94. 94.
    Nagayama, K., Mori, Y., Shimada, K.: Shock Hugoniot compression curve for water up to 1 GPa by using a compressed gas gun. J. Appl. Phys. 91, 476–482 (2002)CrossRefGoogle Scholar
  95. 95.
    Bergant, A.: Developments in unsteady pipe flow friction modeling. J. Hydraul. Res. 39, 249–257 (2001)CrossRefGoogle Scholar
  96. 96.
    Suo, L., Wylie, E.B.: Complex wavespeed and hydraulic transients in viscoelastic pipes. J. Fluids Eng. 112, 496–500 (1990)CrossRefGoogle Scholar
  97. 97.
    Leighton, T.G.: The Acoustic Bubble. Academic Press (1994)Google Scholar
  98. 98.
    Matsumoto, Y., Yoshizawa, S.: Behaviour of a bubble cluster in an ultrasound field. Int. J. Numer. Meth. Fluids 47, 591–601 (2005)MATHCrossRefGoogle Scholar
  99. 99.
    Shimada, M., Matsumoto, Y., Kobayashi, T.: Influence of the nuclei size distribution on the collapsing behavior of the cloud cavitation. JSME Int. J. Ser. B 43, 380–385 (2000)CrossRefGoogle Scholar
  100. 100.
    Wang, Y.-C., Brennen, C.E.: Numerical computation of shock waves in a spherical cloud of cavitation bubbles. J. Fluids Eng. 121, 872–880 (1999)CrossRefGoogle Scholar
  101. 101.
    Wang, Y.-C.: Effects of nuclei size distribution on the dynamics of a spherical cloud of cavitation bubbles. J. Fluids Eng. 121, 881–886 (1999)CrossRefGoogle Scholar
  102. 102.
    Delale, C.F., Schnerr, G.H., Sauer, J.: Quasi-one-dimensional steady-state cavitating nozzle flows. J. Fluid Mech. 427, 167–204 (2001)MATHCrossRefGoogle Scholar
  103. 103.
    Delale, C.F.: Thermal damping in cavitating nozzle flows. J. Fluids Eng. 124, 969–976 (2002)CrossRefGoogle Scholar
  104. 104.
    Delale, C.F., Okita, K., Matsumoto, Y.: Steady-state cavitating nozzle flows with nucleation. J. Fluids Eng. 127, 770–777 (2005)CrossRefGoogle Scholar
  105. 105.
    Wang, Y.-C., Brennen, C.E.: One-dimensional bubbly cavitating flows through a converging-diverging nozzle. J. Fluids Eng. 120, 166–170 (1998)CrossRefGoogle Scholar
  106. 106.
    Wang, Y.-C.: Stability analysis of one-dimensional steady cavitating nozzle flows with bubble size distribution. J. Fluids Eng. 122, 425–430 (2000)CrossRefGoogle Scholar
  107. 107.
    An, Y.: Formulation of multibubble cavitation. Phys. Rev. E 83, Art. No. 066313 (2011)Google Scholar
  108. 108.
    Bremond, N., Arora, M., Ohl, C.D., Lohse, D.: Controlled muntibubble surface cavitation. Phys. Rev. Lett. 96, Art. No. 224501 (2006)Google Scholar
  109. 109.
    Ilinskii, Y.A., Hamilton, M.F., Zabolotskaya, E.A.: Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics. J. Acoust. Soc. Am. 121, 786–795 (2007)CrossRefGoogle Scholar
  110. 110.
    Ida, M.: Bubble-bubble interaction: A potential source of cavitation noise. Phys. Rev. E 79, Art. No. 016307 (2009)Google Scholar
  111. 111.
    Ando, K.: Effects of polydispersity in bubbly flows. PhD Thesis. California Institute of Technology (2010), http://thesis.library.caltech.edu/5859/

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Keita Ando
    • 1
  • Tim Colonius
    • 2
  • Christopher E. Brennen
    • 2
  1. 1.Keio UniversityTokyoJapan
  2. 2.California Institute of TechnologyPasadenaUSA

Personalised recommendations